The local spectrum of the Dirac operator for the universal cover of SL2(R)

Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of SL2(R), exhibiting it as the generator of KK1(C,A), where A is the reduced C⁎-algebra of the group. This yields a new and direct computation of the K-theory of A. A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group. We provide a detailed analysis of the localised spectra of the Dirac operator and compute the Dirac cohomology.